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II. Constitutive Relations and Intrinsic Properties

Whenever one deals with materials the notion coming first to one’s mind is that of constitutive law. In broad terms, this is a relationship between stresses and strains. As our representations of these concepts is by matrices, a constitutive law is simply a mathematical relation - algebraic or analytic - between two 3×3 matrices. If we denote by σ the matrix of stresses and by ε the matrix of strains, then a constitutive law is a relation of the form σ = Σ(ε) where the function Σ is accessible to evaluation. At this point we must emphasize, once and for all, that in an experiment σ is the applied stress, while ε is the resulting strain. Then Σ implicitly contains the influence of material. Indeed every action of ours is done by forces; in other words we cannot control but forces and, if anything else, through forces. This tiny detail seems to have no noticeable theoretical consequences all along the history of Mechanics. It is seldom noticed that in the framework of Analytical Mechanics there are no uncontrollable quantities. These however became everyday events, if we may say so, mostly as a consequence of technological achievements. In the case of materials we have always uncontrollable events, but Continuum Mechanics – the science of this resort – seems to operate under the very same classical idea that all the internal motions of the bodies are controllable. Or, if they are not controllable, their effect can at least be wiped out in the process of measurement. By this working philosophy, Continuum Mechanics is akin the kinetic theory of gases. In what follows we assume that Σ is analytic, an assumption that ultimately comes to a representation of stresses in terms of strain matrix by a formal series. This analytic nature of the constitutive law is strongly suggested by experiment. The novelty we exploit here is that everything comes down to the fact that the uncontrollable features in the relationship between stresses and strains are quantifiable in a special way. This seems to be the reason why the intellect has been induced in creating certain structures.

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